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I could not find any lower bound on the diamond norm for two uniformly random unitaries of dimension D sampled from the haar measure.

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This answer won't actually give you a bound, but will provide some information that may help you in your search. You may be able to find an answer in the random matrix theory literature if you translate the question into different terms, as I will describe.

First, suppose that $\Phi_0(X) = U_0 X U_0^{\dagger}$ and $\Phi_1(X) = U_1 X U_1^{\dagger}$ are any two unitary channels. In this special case, the diamond norm distance between these two channels can be expressed as follows: $$ \|\Phi_0 - \Phi_1\|_{\diamond} = 2 \sqrt{1 - \delta(U_0^{\dagger} U_1)^2}, $$ where $\delta(A)$ denotes the minimum absolute value taken over the numerical range of a given operator $A$. That is, $$ \delta(A) = \min_{|\psi\rangle} |\langle \psi | A | \psi\rangle|, $$ where the minimum is over all unit vectors $|\psi\rangle$.

Now, if you're interested in $U_0$ and $U_1$ being Haar-random, then you might as well set $U_0 = I$ and $U_1 = U$ for $U$ Haar-random, by the fact that the diamond norm is unitarily invariant. To understand the distribution of the diamond norm distance, you therefore need to understand something about the numerical range of a Haar-random unitary. Because unitary operators are normal, their numerical range is the convex hull of their eigenvalues, so you're essentially trying to understand something about the eigenvalues of a Haar-random unitary. This topic has been studied extensively in random matrix theory.

I do not know enough about this topic to give you precise bounds, but it is clear that as the dimension $D$ grows, the probability that the diamond norm distance equals the maximum possible value 2 (which means the two channels can be discriminated perfectly) approaches 1. Indeed, in order to have diamond norm distance strictly less than 2, all of the eigenvalues of $U$ (or $U_0^{\dagger} U_1$) must fall within an arc on the unit circle having length strictly less than $\pi$, which is extremely unlikely for large $D$ (surely the probability is exponentially small in $D$).

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